Having the Queen pretending to be the leader of the Zygons, somehow knowing exactly what their plan was, and pulling it off well enough to fool the other Zygons.
As cool as it was to have all thirteen Doctors join forces to save Gallifrey, some have questioned the credibility of the scene. How in the hell did they manage to summon all thirteen Doctors at once, and got them to agree to come over? Did Eleven travel back and visit each of them himself, or did the Moment bring them altogether? And what about the major repercussions of crossing his own timeline? Given that three Doctors was catastrophic enough already, having all thirteen together alone would destroy reality. Or repair it. (The episode, however, establishes that past Doctors are unable to retain the memories of interacting with their future selves, which also offers a retroactive fix-it for all past multi-Doctor stories going back to 1972-73.
Billie Piper as The Moment's body. Some don't mind it, others... still have issues over Rose. Although, there were some who disliked Rose as a character but thought Piper as The Moment was fantastic. And, technically speaking, The Moment wasn't actually Rose, but the Bad Wolf entity (which possessed Rose). Then there were people who were unhappy Rose wasn't in the special, feeling they should have had her and Doctor 10.5.
The idea of the War Doctor. Some say John Hurt was fantastic in the role, others believe that the character could've been filled with either Paul McGann (as number 8) or Christopher Eccleston (as number 9). While both Doctors do appear in the story in brief but crucial roles, it's only through stock footage, and the former already made an appearance in a lead-up mini-sode.
Moreover, Eccleston declined to return for the special. Thus, we never get to dwell on his Doctor finding out he never really destroyed his own planet and people, though he's going to forget anyway. It would have been fitting as he's the first Doctor shown to bear that guilt ever since the show was revived.
The Ninth Doctor was also implied to have recently regenerated way back in Series 1 (surprised to see his big ears), so it was naturally assumed the Eighth Doctor ended the Time War. Regardless, Moffat was having trouble picturing the Eighth Doctor fighting in a war because of what his character was like, even when Eight started turning darker. Which led to a prequel mini-episode with a regeneration scene for his Doctor to explain away the War Doctor's existence, so it's still a win in the end for fans who wanted to see Paul McGann again, along with making his Big Finish Doctor Who adventures canon. And John Hurt's Doctor even goes out on a line that links back to Nine's ears comment.
Clara suddenly being able to close the TARDIS' doors with a snap of her fingers), doing cool stunts on her motorcycle and managing to talk the Doctor down from burning Gallifrey again. Audience reactions ranged from "Wow!" to "What the hell?!" She's becoming very divisive in the fanbase. Part of the issue is that the finger snapping to close the door was a big deal for the Doctor to do, so for Clara to just casually do it is a bit off-putting. That and it marks the fact that the TARDIS has apparently done a 180 on its opinion of her (it had previously blatantly hated her) because the paradoxes surrounding Clara have been resolved! It's probably also because Clara (i.e. the Gallifreyian version of her) introduced the Doctor to his TARDIS. Though that brings up it's own issues since the TARDIS explicitly stated it chose the Doctor.
Some fans of the classic Who felt the special was more of a celebration of the new era rather than a celebration of all of Who history outside a few cameos and the fact that the event that separates old from new who is the focus of the plot. Fans who joined in during the reboot pointed out that the Big Finish Audio Drama, "Night of the Doctor" and The Five(ish) Doctors Reboot were all more focused on Classic Who. On the other hand, the cameo of Tom Baker, plus the quips of the War Doctor at the behaviors of Ten and Eleven were great big nods at the classic roots. Although for classic series fans may still point out that a webisode and a digital-exclusive parody, while both enjoyable and welcome, aren't exactly on the same level as a major international TV-and-cinema release, and both the Curator and the War Doctor are still entirely new versions of the Doctor introduced in the new series (classic series fans, for example, argue that there was no real reason to create a whole new Doctor to represent the 'classic' series when an actual classic series Doctor could easily have filled that role just fine).
The reveal that the Doctors saved Gallifrey instead of destroying it. Some thought it was perfectly in character for the Doctor to Take a Third Option instead of needlessly killing millions of innocent lives. Species and characters who seem to be Killed Off for Real only to be Not Quite Dead or Back from the Dead have always been a staple of the series since day one. After all if the Daleks, Cybermen, Davros, the Master (and the planet Skaro) can keep coming back after being destroyed time and time again, why not the Time Lords and Gallifrey? Still diehard Russell T. Davies fans were furious at what they saw as Moffat needlessly tampering with what they saw as an essential part of the Doctor's backstory (for the new series at least).
The hopeful ending and Eleven's monologue about returning to Gallifrey "the long way 'round" is a lot more bittersweet, if not bitter, thanks to later events.
Eleven never makes it home, but rather spends a millennium defending Trenzalore and ensuring Gallifrey cannot return to the main universe in the very next episode, which ends with him regenerating.
The three-part finale of Series 9 reveals that Gallifrey does return to the main universe at a much later point in its lifespan...but also sees the Time Lord powers-that-be betray the Twelfth Doctor, arranging for his capture and inadvertently paving the way for Clara Oswald's death. From there, the Doctor is Driven to Madness with grief and rage and fights his way out of his torture chamber he's imprisoned in — and back to Gallifrey — over four-and-a-half billion years. There, he is almost executed on Rassilon's orders, but instead is able to bloodlessly overthrow him thanks to his reputation as the man who ended the Time War. And then he chooses to forsake his homeworld and people in a Batman Gambit to save Clara from the grave, even shooting the General and forcing him/her to regenerate to make a getaway with her, feeling he is owed her revival after everything he's done for Gallifrey and the universe — even as his acts risk all space and time. In the end, he is once again a renegade from his homeworld, on the run from his people, and he loses Clara and many of his key physical/emotional memories of her for good as well by way of returning to his best self. Poor Doctor...
Heartwarming in Hindsight: The Face of Boe's message in "Gridlock" that, "You are not alone," can take on a much more positive meaning with this episode, since it can be interpreted now as a message that Gallifrey is safe.
Hollywood Homely: Osgood. The Zygon version notes how jealous Osgood is of her prettier sister, but the actress playing her is a pretty actress in nerd garb.
Narm: The repeated shots of the children of Gallifrey to some, who felt it bordered on glurge.
Ship Tease: Clara and the Eleventh Doctor act like a couple at the start of the episode, even discussing going on a date for cocktails on Future Mars (though the actual word isn't used). At the end of the episode, she also kisses the Doctor on the cheek and strokes his face.
Sadly averted with regards to Ten and Rose, due to the fact that Billie Piper does not appear as Rose (though she is still credited under this character name), but rather a WMD disguised as Rose Tyler. Other than a quick reference, there is no opportunity for any revisiting of the ship tease between those two characters.
They Wasted a Perfectly Good Plot: How they got all 13 Doctors together would have been a fun story. And as of the ending of Series 9, how and when Twelve was able to participate still hasn't been revealed. He's only able to exist because the Time Lords give the dying Eleven a new cycle of regenerations in the next story, so it involves a Stable Time Loop of some kind. And presumably Clara can't be with him at the time. Given what the Time Lords do to Twelve and by extension Clara in "Face the Raven" / "Heaven Sent"/ "Hell Bent", the fact that where this special falls in his time line hasn't yet been revealed begs the question: If the Twelfth Doctor is summoned after that story unfolds, why does he save the people responsible for his betrayal and torment? Is it just to keep the time loop intact, or does his still genuinely care for them despite their evil? Or did he save Gallifrey solely for the sake of the ordinary Gallifreyans, and the Time Lord elite just happened to get dragged along for the ride?
The "13 Doctors" issue is at least somewhat justified in that, of the thirteen actors to play the Doctor on TV, at the time of recording three were no longer with us, one was in his early 80s (and thus likely would not have been able to participate much more than what we got), one had made clear his lack of interest in participating and several of the others had all also aged quite significantly since appearing on the show. This meant that that a true "The Thirteen Doctors" would have been quite difficult to pull off in a way which pleased everyone to begin with.
As mentioned above under Broken Base, some fans view the whole special as one of these with regards to the classic series, as outside of background references (several of which are blink-and-you-miss-them), stock footage and a handful of cameo appearances, the special as a whole tends to focus on plot points and developments introduced in the new series. To some critics, this reaches the point where it may seem like more of a celebration of the new series than the show as a whole. As an example, while the War Doctor was generally well-received several of these critics have argued that his role in events (as the Doctor who fought in the Time War) could have easily have been filled by the Eighth Doctor, and that there was arguably no real reason to create an all-new Doctor to 'represent' the classic series rather than using an actual classic series Doctor (even if, as mentioned above, using all of them might have been impractical).
Visual Effects of Awesome: Clara driving her motorcycle into the TARDIS, in one continuous shot. You'd swear the thing really is bigger on the inside. Although the transition had actually been introduced earlier (in "The Snowmen") this upped the ante by incorporating a moving vehicle.